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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Autumn Mathematical Competition
2008 Bulgarian Autumn Math Competition
2008 Bulgarian Autumn Math Competition
Part of
Bulgarian Autumn Mathematical Competition
Subcontests
(20)
Problem 12.4
1
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Set of numbers doesn't change under arithmetic operation
Veni writes down finitely many real numbers (possibly one), squares them, and then subtracts 1 from each of them and gets the same set of numbers as in the beginning. What were the starting numbers?
Problem 12.3
1
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Functional equation in reals
Find all continuous functions
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
such that (f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y) \forall x,y\in \mathbb{R}
Problem 12.2
1
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Strange collinearity involving the incenter
Let
A
B
C
ABC
A
BC
be a triangle, such that the midpoint of
A
B
AB
A
B
, the incenter and the touchpoint of the excircle opposite
A
A
A
with
A
C
‾
\overline{AC}
A
C
are collinear. Find
A
B
AB
A
B
and
B
C
BC
BC
if
A
C
=
3
AC=3
A
C
=
3
and
∠
A
B
C
=
6
0
∘
\angle ABC=60^{\circ}
∠
A
BC
=
6
0
∘
.
Problem 12.1
1
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System of inequalities with logs and a parameter
Determine the values of the real parameter
a
a
a
, such that the solutions of the system of inequalities
{
log
1
3
(
3
x
−
6
a
)
+
2
log
a
3
<
x
−
3
log
1
3
(
3
x
−
18
)
>
x
−
5
\begin{cases} \log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\ \log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\ \end{cases}
{
lo
g
3
1
(
3
x
−
6
a
)
+
l
o
g
a
3
2
<
x
−
3
lo
g
3
1
(
3
x
−
18
)
>
x
−
5
form an interval of length
1
3
\frac{1}{3}
3
1
.
Problem 11.4
1
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Square free integers in floor function
a) Prove that
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
is odd iff
⌊
2
{
x
2
}
⌋
=
1
\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1
⌊
2
{
2
x
}
⌋
=
1
(
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denotes the largest integer less than or equal to
x
x
x
and
{
x
}
=
x
−
⌊
x
⌋
\{x\}=x-\lfloor x\rfloor
{
x
}
=
x
−
⌊
x
⌋
). b) Let
n
n
n
be a natural number. Find the number of square free numbers
a
a
a
, such that
⌊
n
a
⌋
\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor
⌊
a
n
⌋
is odd. (A natural number is square free if it's not divisible by any square of a prime number).
Problem 11.3
1
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Coloring and intersection of diagonals in a polygon
In a convex
2008
2008
2008
-gon some of the diagonals are coloured red and the rest blue, so that every vertex is an endpoint of a red diagonal and no three red diagonals concur at a point. It's known that every blue diagonal is intersected by a red diagonal in an interior point. Find the minimal number of intersections of red diagonals.
Problem 11.2
1
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Equal areas in a right triangle
On the sides
A
B
AB
A
B
and
A
C
AC
A
C
of the right
△
A
B
C
\triangle ABC
△
A
BC
(
∠
A
=
9
0
∘
\angle A=90^{\circ}
∠
A
=
9
0
∘
) are chosen points
C
1
C_{1}
C
1
and
B
1
B_{1}
B
1
respectively. Prove that if
M
=
C
C
1
∩
B
B
1
M=CC_{1}\cap BB_{1}
M
=
C
C
1
∩
B
B
1
and
A
C
1
=
A
B
1
=
A
M
AC_{1}=AB_{1}=AM
A
C
1
=
A
B
1
=
A
M
, then
[
A
B
1
M
C
1
]
+
[
A
B
1
C
1
]
=
[
B
M
C
]
[AB_{1}MC_{1}]+[AB_{1}C_{1}]=[BMC]
[
A
B
1
M
C
1
]
+
[
A
B
1
C
1
]
=
[
BMC
]
.
Problem 11.1
1
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Arithmetic sequence satisfies special property
Let
a
1
,
a
2
,
…
a_{1},a_{2},\ldots
a
1
,
a
2
,
…
be an infinite arithmetic progression. It's known that there exist positive integers
p
,
q
,
t
p,q,t
p
,
q
,
t
such that
a
p
+
t
p
=
a
q
+
t
q
a_{p}+tp=a_{q}+tq
a
p
+
tp
=
a
q
+
tq
. If
a
t
=
t
a_{t}=t
a
t
=
t
and the sum of the first
t
t
t
numbers in the sequence is
18
18
18
, determine
a
2008
a_{2008}
a
2008
.
Problem 10.4
1
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Cute graph problem
There are
3
≤
n
≤
25
3\leq n\leq 25
3
≤
n
≤
25
passengers in a bus some of which are friends. Every passenger has exactly
k
k
k
friends among the passengers, no two friends have a common friend and every two people, who are not friends have a common friend. Find
n
n
n
.
Problem 10.3
1
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Diophantine equation
Find all natural numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
, such that
7
x
+
1
3
y
=
2
z
7^{x}+13^{y}=2^{z}
7
x
+
1
3
y
=
2
z
.
Problem 10.2
1
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Equal areas condition
Let
△
A
B
C
\triangle ABC
△
A
BC
have
M
M
M
as the midpoint of
B
C
BC
BC
and let
P
P
P
and
Q
Q
Q
be the feet of the altitudes from
M
M
M
to
A
B
AB
A
B
and
A
C
AC
A
C
respectively. Find
∠
B
A
C
\angle BAC
∠
B
A
C
if
[
M
P
Q
]
=
1
4
[
A
B
C
]
[MPQ]=\frac{1}{4}[ABC]
[
MPQ
]
=
4
1
[
A
BC
]
and
P
P
P
and
Q
Q
Q
lie on the segments
A
B
AB
A
B
and
A
C
AC
A
C
.
Problem 10.1
1
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Parameter equation with 3 real roots
For which values of the parameter
a
a
a
does the equation
(
2
x
−
a
)
a
x
2
−
(
a
2
+
a
+
2
)
x
+
2
(
a
+
1
)
=
0
(2x-a)\sqrt{ax^2-(a^2+a+2)x+2(a+1)}=0
(
2
x
−
a
)
a
x
2
−
(
a
2
+
a
+
2
)
x
+
2
(
a
+
1
)
=
0
has three different real roots.
Problem 9.4
1
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A Doable Nikolai problem
Stoyan and Nikolai have two
100
×
100
100\times 100
100
×
100
chess boards. Both of them number each cell with the numbers
1
1
1
to
10000
10000
10000
in some way. Is it possible that for every two numbers
a
a
a
and
b
b
b
, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)?Nikolai Beluhov
Problem 9.3
1
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Divisors of polynomials
Let
n
n
n
be a natural number. Prove that if
n
5
+
n
4
+
1
n^5+n^4+1
n
5
+
n
4
+
1
has
6
6
6
divisors then
n
3
−
n
+
1
n^3-n+1
n
3
−
n
+
1
is a square of an integer.
Problem 9.2
1
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Touching circles imply radius ratio
Given a
△
A
B
C
\triangle ABC
△
A
BC
and the altitude
C
H
CH
C
H
(
H
H
H
lies on the segment
A
B
AB
A
B
) and let
M
M
M
be the midpoint of
A
C
AC
A
C
. Prove that if the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
,
k
k
k
and the circumcircle of
△
M
H
C
\triangle MHC
△
M
H
C
,
k
1
k_{1}
k
1
touch, then the radius of
k
k
k
is twice the radius of
k
1
k_{1}
k
1
.
Problem 9.1
1
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System of equations
Solve the system
{
x
2
y
2
+
∣
x
y
∣
=
4
9
x
y
+
1
=
x
+
y
2
\begin{cases} x^2y^2+|xy|=\frac{4}{9}\\ xy+1=x+y^2\\ \end{cases}
{
x
2
y
2
+
∣
x
y
∣
=
9
4
x
y
+
1
=
x
+
y
2
Problem 8.4
1
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Maximum number of obtuse angles
Let
M
M
M
be a set of
99
99
99
different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of
M
M
M
inside in. What is the maximum number of obtuse angles formed by two rays in
M
M
M
?
Problem 8.3
1
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Sum of digits of a prime
Prove that there exists a prime number
p
p
p
, such that the sum of digits of
p
p
p
is a composite odd integer. Find the smallest such
p
p
p
.
Problem 8.2
1
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Geometry problem with angle bisectors
Let
△
A
B
C
\triangle ABC
△
A
BC
have
∠
A
=
2
0
∘
\angle A=20^{\circ}
∠
A
=
2
0
∘
and
∠
C
=
4
0
∘
\angle C=40^{\circ}
∠
C
=
4
0
∘
. We've constructed the angle bisector
A
L
AL
A
L
(
L
∈
B
C
L\in BC
L
∈
BC
) and the external angle bisector
C
N
CN
CN
(
N
∈
A
B
N\in AB
N
∈
A
B
). Find
∠
C
L
N
\angle CLN
∠
C
L
N
.
Problem 8.1
1
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Parametric equality
Solve the equation
∣
x
−
m
∣
+
∣
x
+
m
∣
=
x
|x-m|+|x+m|=x
∣
x
−
m
∣
+
∣
x
+
m
∣
=
x
depending on the value of the parameter
m
m
m
.